At which location will the weight of an object be minimum?

Introduction / Context:
This question explores how gravitational force varies with position relative to the Earth. It is common in basic physics and general science to ask where on or inside the Earth a body would weigh the most or the least. The options include points on the surface such as the poles and the equator, as well as the Earth's centre. Knowing the correct location where weight is minimum requires understanding both gravity inside a sphere and the effects of rotation and shape.

Given Data / Assumptions:

• We consider weight W = m * g, where m is constant and g varies with position.
• Locations given: North Pole, South Pole, Equator, centre of the Earth, and a high mountain peak.
• The Earth is approximated as a sphere with uniform or smoothly varying density.
• Rotational effects act mainly at the surface, especially at the equator.

Concept / Approach:
On the Earth's surface, g is not exactly the same everywhere. It is slightly higher at the poles and slightly lower at the equator due to the Earth's rotation and its equatorial bulge. As you go above the surface, g decreases with height. Inside a uniform sphere, g decreases linearly with decreasing radius from the centre. At the very centre of the Earth, the gravitational pulls from all directions cancel, making net g equal to zero. Since weight is m * g, the minimum possible weight (actually zero) occurs at the centre of the Earth.

Step-by-Step Solution:
Step 1: Recognise that weight depends directly on the local value of g. Step 2: At the poles, g is slightly larger than at the equator, so weight is slightly higher, not lower. Step 3: At the equator, rotation reduces effective g a little, so weight is less than at the poles but still not zero. Step 4: On high mountain peaks, the distance from the Earth's centre is slightly larger, so g decreases a bit, but again not to zero. Step 5: At the centre of the Earth, gravitational forces from all directions balance and g becomes zero, making weight W = m * 0 = 0, which is the minimum possible value.

Verification / Alternative check:
We can compare the magnitude of g at various locations. Standard values show g ≈ 9.83 m/s^2 at the poles and ≈ 9.78 m/s^2 at the equator. On high mountains it is slightly less than at sea level but still close to 9.7–9.8 m/s^2. The theory of gravity inside a sphere states that g is proportional to r (distance from the centre), and at r = 0, g = 0. This mathematical relationship confirms that the centre of the Earth is the unique location where weight becomes zero and hence is minimum.

Why Other Options Are Wrong:
The North Pole: g is actually slightly higher here than at the equator, so weight is not minimum. The South Pole: Same reasoning as the North Pole; weight is slightly higher than at the equator. The Equator: Weight is lower than at the poles but still not the minimum possible because g is not zero. On a high mountain peak: g decreases with height but does not reach zero on any ordinary mountain, so weight is reduced but not minimum.

Common Pitfalls:
The most common mistake is to choose the equator, because students remember that g is smallest there on the surface due to centrifugal effects. However, the question is about the absolute minimum weight anywhere, not just among surface points. Another pitfall is to ignore the possibility of going inside the Earth altogether. Remembering that g becomes zero at the centre, giving zero weight, is crucial to arriving at the correct answer.

Final Answer:
The weight of an object is minimum (actually zero) at the center of the Earth.